Lemma 5.2 ([1]},[2]}) 
Let \(T\)  be a contraction on a Hilbert space \(\mathcal {H}\)  for which there exists a Hilbert space \(\mathcal {E}\) , \(\Psi \in \mathcal {S}\mathcal {A}_n(\mathcal {E},\mathcal {E})\)  and an isometry \(\Pi :\mathcal {H}\rightarrow H_{\mathcal {E}}^{2}(n)\)  (where, \(n \ge 1\) ), such that \(\Pi T^* = M_{\Psi }^* \Pi \)  and \(H_{\mathcal {E}}^{2}(n) = \bigvee _{\mathbf {k} \in \mathbb {N}^n} M_{\mathbf {z}}^{\mathbf {k}} \Pi \mathcal {H}\) . If \(T\)  is a pure contraction on \(\mathcal {H}\) , then \(\Psi (0)\)  is a c.n.u. contraction on \(\mathcal {K}\) .

